Scaling limits and critical phenomena in interacting particle systems

Abstract

This thesis studies three different large-scale phenomena in statistical mechanics. The first phenomenon studied is superdiffusivity in two different models, a diffusion in a random environment, and a critical stochastic partial differential equation in Chapters 2 and 3 respectively. Both models are diffusive systems, which are perturbed by some external forcing. The effect of this forcing can be measured using the so-called Diffusion coefficient D(t). In this part of the Thesis it is proven that D(t) diverges like (logt)^1/2 and (logt)^2/3, respectively, up to Tauberian inversions. These results prove conjectures made for the corresponding models. The proofs use tools from Gaussian Analysis and an iterative estimation scheme to study the resolvent of the generator of the process.The second phenomenon is a near-critical limit of a conformally invariant model, namely the dimer model in Chapter 4. The fluctuations of the planar dimer model in two dimensions are one of the few models from statistical mechanics where conformal invariance has been rigorously proven. This conformal invariance holds for certain critical weights and certain boundary conditions. In this chapter, we study the dimer model near criticality. We execute part of the program initiated in [N. Makarov and S. Smirnov, Off-critical lattice models and massive SLEs, 2009, Proceedings of ICMP 2009], by finding a scaling limit for the corresponding height functions, and connecting this scaling limit to massive SLE2. As is typical for near-critical models, this limit is no longer conformally invariant but conformally covariant. The proof uses a connection to loop-erased random walks via Temperley’s bijection and Wilson’s algorithm. We also prove an exact discrete Girsanov identity for the triangular lattice, which might be of independent interest.The third phenomenon is the almost sure convergence of the asymptotic speed of a second-class particle in an interacting particle system started from specific non-stationary initial conditions. In particular, we study the stochastic six-vertex model on the quadrant Z≥0 ×Z≥0 with step initial conditions, i.e. every incoming edge from the left is occupied by a particle and every incoming edge from the bottom is unoccupied. We then add a single second-class particle coming in from below. The main theorem of Chapter 5 states that the speed X_T/T of this second-class particle converges almost surely to a random limit. This allows one to define the stochastic six-vertex speed process. We use tools from integrable probability to obtain precise bounds on the fluctuations of the height functions around its limit shape together with a novel result that allows us to control the behavior of an individual second-class particle by controlling the behavior of a larger number of third-class particles.